General Equilibrium Oligopoly and Ownership Structure∗

JOSÉ AZAR XAVIER VIVESIESE Business School IESE Business School

October 5, 2020

Abstract

We develop a tractable general equilibrium framework in which firms are large and have marketpower with respect to both products and labor, and in which a firm’s decisions are affected by itsownership structure. We characterize the Cournot–Walras equilibrium of an economy where eachfirm maximizes a share-weighted average of shareholder utilities—rendering the equilibrium inde-pendent of price normalization. In a one-sector economy, if returns to scale are non-increasing thenan increase in “effective” market concentration (which accounts for common ownership) leads to de-clines in employment, real wages, and the labor share. Yet when there are multiple sectors, due toan intersectoral pecuniary externality, an increase in common ownership could stimulate the econ-omy when the elasticity of labor supply is high relative to the elasticity of substitution in productmarkets. We characterize for which ownership structures the monopolistically competitive limit oran oligopolistic one are attained as the number of sectors in the economy increases. When firms haveheterogeneous constant returns to scale technologies we find that an increase in common ownershipleads to markets that are more concentrated.

Keywords: common ownership, portfolio diversification, macro economy, corporate governance, laborshare, market power, oligopsony, antitrust policy

JEL Classification: D43, D51, E11, L21, L41

∗This paper subsumes results in our paper entitled “Oligopoly, Macroeconomic Performance, and Competition Policy” andwas the basis of the Walras–Bowley lecture of Vives. We are grateful to two anonymous referees for useful suggestions thatimproved the paper’s presentation. For helpful comments we thank Daron Acemoglu and Nicolas Schutz as well as partici-pants in seminars at the ECB, UC Davis, the University of Chicago, Northwestern, NYU Stern, and Yale and in workshops atthe Bank of England, CEMFI, CRESSE, HKUST, Korea University, Mannheim, and the SCE II Congress (Barcelona). GiorgiaTrupia and Orestis Vravosinos provided excellent research assistance. Vives’s interest in this paper’s topic was spurred byearly (mid-1970s) conversations with Josep Ma. Vegara and Joaquim Silvestre. We gratefully acknowledge financial supportfor Azar by Secretaria d’Universitats I Recerca, Generalitat de Catalunya (Ref. 2016 BP00358), for Vives by the European Re-search Council (Advanced Grant no. 789013), and for both by the Ministry of Science, Innovation and Universities with projectPGC2018-096325-B-I00 (MCIU/AEI/FEDER, UE).

CMonnerTypewritten TextForthcoming at Econometrica

1 Introduction

Oligopoly is widespread and allegedly on the rise. Many industries are characterized by oligopolisticconditions—including, but not limited to, the digital ones dominated by GAFAM: Google (now Alpha-bet), Apple, Facebook, Amazon, and Microsoft. These firms, as well as others, have influence in theaggregate economy.1 Yet oligopoly is seldom considered by macroeconomic models, which focus onmonopolistic competition because of its analytical tractability. A typical limitation of monopolistic com-petition models is that they have no role for market concentration to play in conditioning competitionbecause the summary statistic for competition is the elasticity of substitution. In the field of internationaltrade, a few papers consider oligopoly—but with a continuum of sectors and hence with negligible firmsin relation to the economy (Neary, 2003a,b; Atkeson and Burstein, 2008). Furthermore, all these papersassume that firms maximize profits even with ownership structures which induce a departure fromprofit maximization.

In this paper we build a tractable general equilibrium model of oligopoly allowing for ownership di-versification, characterize its equilibrium and comparative statics properties, and then use it to analyzethe effect of competition policies. Our contribution is mostly methodological, although we have appliedthe multisector version of our model elsewhere to explain the evolution of macroeconomic magnitudesin a calibration exercise (Azar and Vives, 2018, 2019a). We adopt this approach in light of (a) the in-creasing concentration in the US economy with respect to both product and labor markets and (b) theincreasing extent of common ownership due to the increase in institutional investment—especially inindex funds (thus, for almost 90% of S&P 500 firms, the largest proportion of shares is held by the “Big 3”asset managers: BlackRock, Vanguard, and State Street). These trends have raised concerns of increasedmarket power and markups (Azar, 2012; Azar, Schmalz, and Tecu, 2018; De Loecker, Eeckhout, andUnger, 2020) as well as calls for antitrust action and regulation of common ownership, topics that arehotly debated (see e.g. Elhauge, 2016; Posner, Scott Morton, and Weyl, 2017).

The difficulties of incorporating oligopoly into a general equilibrium framework have hindered themodeling of market power in macroeconomics and international trade. The reason is that there is nosimple objective for the firm when firms are not price takers.2 In a general equilibrium, moreover, firmswith pricing power will affect not only their own respective profits but also the wealth of consumersand therefore demand (these feedback effects are sometimes referred to as “Ford effects”). Firms that arelarge relative to factor markets also have to take into account their impact on factor prices. Gabszewiczand Vial (1972) proposed the Cournot-Walras equilibrium concept assuming firms maximize profit ingeneral equilibrium oligopoly but then equilibrium depends on the choice of numéraire.3 This problem

1In 2019, GAFAM accounted for nearly 15% of US market capitalization. An extreme example is provided by Samsung andHyundai, which are large relative to Korea’s economy (Gabaix, 2011). In the United States, General Motors and Walmart—despite never employing more than 1% of the country’s workforce—often figure prominently in local labor markets.

2With price-taking firms, a firm’s shareholders agree unanimously that the objective of the firm should be to maximize itsown profits. This result is known as the Fisher separation theorem (DeAngelo, 1981), which Hart (1979) extends to incompletemarkets. In fact, Arrow’s impossibility result on preference aggregation was derived precisely when attempting to generalizethe theory of the firm with multiple owners (see Arrow, 1984).

3When firms have market power, the outcome of their optimization depends on what price is taken as the numéraire since

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has been sidestepped by assuming that there is only one good (an outside good or numéraire;that own-ers of the firm care about see e.g. Mas-Colell, 1982) or that firms are small relative to the economy—be itin monopolistic competition (Hart, 1983) or sector oligopoly (Neary, 2003a).

Furthermore, a question arises as to what is the objective of the firm when there is overlapping own-ership due to owners’ diversification. If a firms’ shareholders have holdings in competing firms, theywould benefit from high prices through their effect not only on their own profits, but also on the prof-its of rival firms, as well as internalizing other externalities between firms (Gordon, 1990; Hansen andLott, 1996). Rotemberg (1984) proposes a parsimonious model in which the firm’s manager maximizesa weighted average of shareholders’ utilities and thus internalizes inter-firm externalities.4

We build a model of oligopoly under general equilibrium, allowing firms to be large in relation tothe economy, and then examine the effect of oligopoly on macroeconomic performance. The ownershipstructure allows investors to diversify both intra- and inter-industry. We assume that firms maximize aweighted average of shareholder utilities in Cournot–Walras equilibrium. The weights in a firm’s objec-tive function are given by the influence or “control weight” of each shareholder. This approach solvesthe numéraire problem because indirect utilities depend only on relative prices and not on the choiceof numéraire. Firms are assumed to make strategic decisions that account for the effect of their actionson prices and wages. When making decisions about hiring, for instance, a firm realizes that increasingemployment could result in upward pricing pressure on real wages—reducing not only the firm’s ownprofits but also the profits of all other firms in its shareholders’ portfolios.

We develop first a base model with one sector to present our equilibrium concept and comparativestatic results and then we extend it to a multi-sector economy suitable for calibration. The multi-sectormodel is parsimonious and identifies the key parameters driving equilibrium: elasticity of substitutionacross industries, elasticity of the labor supply, together with the market concentration of each industry,and the ownership structure (i.e., extent of diversification) of investors.

Our approach may shed light on some leading questions. How do output, labor demand, prices,and wages depend on market concentration and the degree of common ownership? To what extentare markups in product markets, and markdowns in the labor market, affected by how much the firminternalizes other firms’ profits? Can common ownership be pro-competitive in a general equilibriumframework? How do common ownership effects change when the number of industries increases? Inthe presence of ownership diversification, is the monopolistically competitive limit (as described by

by changing the numéraire the profit function is generally not a monotone transformation of the original one (see Ginsburgh,1994).

4The maximization of the objective function “weighted average of shareholder utilities” depends on the cardinal propertiesof shareholders’ preferences (violating Arrow’s ordinal postulate). However, it can be microfounded using a purely ordinalmodel—provided shareholder preferences are random from the perspective of the managers who run the firms (Azar, 2012,2017; Brito, Osório, Ribeiro, and Vasconcelos, 2018). Azar (2012) and Brito, Osório, Ribeiro, and Vasconcelos (2018) show that,in a probabilistic voting setting where two managers compete for shareholder votes by developing strategic reputations, thefirm’s objective will be to maximize a weighted average of shareholder utilities without any coordination of the shareholders.It is worth noting that the Big 3 together have stakes that average close to 20% of each publicly traded company in the UnitedStates (Fichtner, Heemskerk, and Garcia-Bernardo, 2017). This gives them enough voting power to be pivotal often. Moreover,Aggarwal, Dahiya, and Prabhala (2019) show that shareholder dissent hurts directors and that director elections matter becauseof career concerns. In particular, these authors show that increasing the votes withheld by only 10% leads to a 24% increase inthe likelihood of director turnover.

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Dixit and Stiglitz, 1977) attained when firms become small relative to the market?—and, more generally,how is that limit affected by ownership structure? Is traditional antitrust policy a complement or rathera substitute with respect to controlling common ownership when the aim is boosting employment?5

In the base model that we develop, there is one good in addition to leisure; also, the model assumesoligopoly in the product market and oligopsony in the labor market. Firms compete by setting theirlabor demands à la Cournot and thus have market power. There is a continuum of risk-neutral own-ers, who each have a proportion of their respective shares invested in one firm and have the balanceinvested in the market portfolio (say, an index fund). This formulation is numéraire-free and allowsus to characterize the equilibrium. The extent to which firms internalize rival firms’ profits dependson market concentration and investor diversification. We demonstrate the existence and uniqueness ofequilibrium, and then characterize its comparative statics properties, while assuming that labor sup-ply is upward sloping (and allowing for some economies of scale in production). The results establishthat, in our model of a one-sector economy, the markdown of real wages with respect to the marginalproduct of labor is driven by the common ownership–modified Herfindahl–Hirschman index (HHI) forthe labor market and also by labor supply elasticity (but not by product market power, since ownershipis proportional to consumption). We perform comparative statics on the equilibrium (employment andreal wages) with respect to market concentration and degree of common ownership, and we developan example featuring Cobb–Douglas firms and consumers with additively separable isoelastic prefer-ences. We find that increased market concentration—due either to fewer firms or to more diversification(common ownership)—depresses the economy by reducing employment, output, real wages, and thelabor share (if one assumes non-increasing returns to scale). When firms have different constant returnsto scale (CRS) technologies, an increase in common ownership leads to a more concentrated market (asmeasured by the HHI) because more efficient firms then gain market share at the expense of weakerrivals. Furthermore, the minimal relative productivity for the least productive firm to be viable is in-creasing in the extent of common ownership.

We extend our base model to allow for multiple sectors, and for differentiated products across sec-tors, with constant elasticity of substitution (CES) aggregators. The firms supplying each industry’sproduct are finite in number and engage in Cournot competition. We allow here for investors to diver-sify both in an intra-industry fund and in an economy-wide index fund. In this extension, a firm de-ciding whether to marginally increase its employment must consider the effect of that increase on threerelative prices: (i) the increase would reduce the relative price of the firm’s own products, (ii) it wouldboost real wages, and (iii) it would increase the relative price of products in other industries (i.e., becauseoverall consumption would increase). This third effect, referred to as inter-sector pecuniary externality, isinternalized only when there is common ownership involving the firm and firms in other industries. Inthis case, the markdown of real wages relative to the marginal product of labor increases with the mod-ified HHI values for the labor market and product markets but decreases with the pecuniary externality(weighted by the extent of competitor profit internalization due to common ownership). We find that

5Azar and Vives (2019b) examine the interaction of competition policy with other government policies to foster employ-ment.

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common ownership always has an anti-competitive effect when increasing intra-industry diversifica-tion but that it can have a pro-competitive effect when increasing economy-wide diversification if theelasticity of labor supply is high in relation to the elasticity of substitution among product varieties. Inthis case, the relative impact of profit internalization on the level of market power in product marketsis higher than in the labor market. It is worth remarking that when the elasticity of labor supply ishigh enough, an increase in economy-wide common ownership always has a pro-competitive effect, nomatter how many sectors the economy has.

We then consider the limiting case when the number of sectors tends to infinity. This formulationallows us to check for whether—and, if so, under what circumstances—the monopolistically compet-itive market of Dixit and Stiglitz (1977) or the oligopolistic ones of Atkeson and Burstein (2008) andNeary (2003a,b) are attained, in the presence of common ownership, when firms become small relativeto the market; it also enables a determination of how ownership structure affects that competitive limit.We find that with incomplete asymptotic diversification, as the number of sectors N in the economygrows, the monopolistically competitive limit is attained if there is either one firm per sector or fullintra-industry common ownership. If full diversification is attained at least as fast as 1/

√N, then profit

internalization is positive in the limit and the Dixit–Stiglitz limit is not attained. We obtain that the limitdegree of profit internalization is increasing in market concentration and in how rapidly diversificationis achieved. The limit markdown may increase or decrease with profit internalization.

Competition policy in the one-sector economy can foster employment and increase real wages byreducing market concentration (with non-increasing returns) and/or the level of diversification (com-mon ownership), which serve as complementary tools. When there are multiple sectors, it is optimal forworker-consumers to have full diversification (common ownership) economy-wide but no extra diver-sification intra-industry—that is, when the elasticity of substitution in product markets is low relativeto the elasticity of labor supply. In this case, competition policy should seek to alter only intra-industryownership structure.

The rest of our paper proceeds as follows. Section 2 describes some further connections with theliterature. Section 3 develops a one-sector model of general equilibrium oligopoly with labor as the onlyfactor of production; this is where we derive comparative statics results with respect to the effects ofmarket concentration on employment, wages, and the labor share. In Section 4 we extend the model toallow for multiple sectors with differentiated products, and we then derive results that characterize thelimit economy as the number of sectors approaches infinity. We also offer some illustrative calibrationsof the model. Section 5 discusses the implications for competition policies, and we conclude in Section 6with a summary and suggestions for further research. Appendix A provides more detail about the caseof increasing returns in production, and the proofs of most results are given in Appendix B.

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2 Connections with the literature

2.1 Theory

Our paper is related to four strands of the literature. The first is the general equilibrium with oligopolyà la the Cournot models of Gabszewicz and Vial (1972), Novshek and Sonnenschein (1978), and Mas-Colell (1982), where the proposed Cournot–Walras equilibrium assumes that firms maximize profits.Here we assume instead that a firm’s manager maximizes a weighted average of shareholder utilitiesand also consider an ownership structure that allows for common ownership.

The second strand encompasses the macroeconomic models with Keynesian features that have in-corporated market power. A precursor of those models is the work by one of Keynes’s contemporaries,Michal Kalecki, on the macroeconomic effects of market power in a two-class economy (Kalecki, 1938,1954). The most closely related papers are perhaps Hart (1982) and d’Aspremont, Ferreira, and Gérard-Varet (1990).6 Hart’s (1982) work differs from ours in assuming that firms are small relative to the overalleconomy and have separate owners. Unions have the labor market power in his model and so equilib-rium real wages are higher than the marginal product of labor; in our model’s equilibrium, real wagesare lower than that marginal product.

In d’Aspremont, Ferreira, and Gérard-Varet (1990), firms are large relative to the economy; however,it is still assumed that firms maximize profits in terms of an arbitrary numéraire and that they competein prices while taking wages as given with an inelastic labor supply. We consider instead the morerealistic case of an elastic labor supply, which yields a positive equilibrium real wage even when marketpower reduces employment to below the competitive level. Our approach differs from theirs also inthat we derive measures of market concentration, discuss competition policy in a general equilibrium,and consider effects on the labor share.7 Furthermore, instead of assuming the existence of consumer-worker-owners (as is typical in the literature), we follow Kalecki (1954) and distinguish between twogroups: worker-consumers and owner-consumers. Our model has a Kaleckian flavor also in relatingproduct market power to the labor share, since in Kalecki (1938), the labor share is determined by theeconomy’s average Lerner index.

The third strand of this literature focuses on international trade models with oligopolistic firms.Neary (2003a) considers a continuum of industries with Cournot competition in each industry, takingthe marginal utility of wealth (instead of the wage) as given. Workers supply labor inelastically andfirms maximize profits. Neary (2003a) finds a negative relationship between the labor share and marketconcentration. Our work differs in that firms are large relative to the economy, and therefore havemarket power in both product and labor markets, and in considering the effects of firms’ ownershipstructure. Neary (2003a) also assumes a perfectly inelastic labor supply, so that changes in market powercan affect neither employment nor output in equilibrium. In contrast, we allow for an increasing labor

6See Silvestre (1993) for a survey of the market power foundations of macroeconomic policy.7Gabaix (2011) also considers firms that are large in relation to the economy but with no strategic interaction among them;

his aim is to demonstrate how microeconomic shocks to large firms can create meaningful aggregate fluctuations. Acemoglu,Carvalho, Ozdaglar, and Tahbaz-Salehi (2012) pursue a similar goal but assume that firms are price takers.

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supply function and examine more potential effects of competition policy. Atkeson and Burstein (2008)also consider a continuum of sectors with Cournot competition in each industry. These authors assumethat goods produced in a country within a sector are better substitutes than across sectors. The aim ofthe paper is to reproduce stylized facts regarding international relative prices.

It is worth noting that in both Atkeson and Burstein (2008) and Neary (2003a), as well as in Dixit andStiglitz (1977), there is a representative household that owns a market portfolio in all the firms. And yet,the firms are assumed to maximize their own profits even though no shareholder would actually wantthis. Thus, there is a tension between the assumed ownership structure and the profit maximizationassumption. The results in Section 4.3, under our assumptions with two classes of agents, in which weconsider the limit as the number of sectors N tends to infinity, make this tension clear. Specifically, withfull asymptotic diversification as N tends to infinity, we obtain the results associated to Dixit-Stiglitz orNeary (2003a) only when there is no rivals’ profit internalization in the limit, and this happens when fulldiversification is attained very slowly (more slowly than 1/

√N).

The fourth strand relates to ownership structure and oligopoly in partial equilibrium. In our modelmanagers internalize the control of the firm by the different owners as in Rotemberg (1984) and O’Brienand Salop (2000), but ours is not a model of the stakeholder corporation as in Magill, Quinzii, and Rochet(2015) since managers only internalize the welfare of owners. The fact that overlapping ownership mayrelax competition was observed by Rubinstein and Yaari (1983) and explored by Reynolds and Snapp(1986), and Bresnahan and Salop (1986). Since overlapping ownership may internalize externalitiesbetween firms, it may have ambiguous welfare effects. Indeed, overlapping ownership may increasemarket power and raise margins yet simultaneously internalize technological spillovers and increaseproductivity (López and Vives, 2019); see He and Huang (2017) for compatible evidence and Geng,Hau, and Lai (2016) for how vertical common ownership links may improve the internalization of patentcomplementarities. Here we will show how common ownership can have pro-competitive effects in amulti-sector economy.8

2.2 Empirics

Our approach may speak about macro trends in the economy in relation to the effects of the evolutionof institutional investment and common ownership patterns, product and labor market concentration,markups and the declining labor share, the consequences for competition and investment, and the im-plications for policy.

The world of dispersed ownership described by Berle and Means (1932) no longer exists in theUnited States. The rise in institutional stock ownership over the past 35 years has been formidable.Pension, mutual, and exchange-traded funds now own the lion’s share of publicly traded US firms. Theasset management industry is concentrated around the three largest managers (BlackRock, Vanguard,and State Street), and there has been a shift from active to passive investors (who are more diversified).

8See Vives (2020) for an exposition of (a) the tension between market power and efficiency as an outcome of commonownership and (b) a parallel with debates in the 1960s and 1970s over the “structure–conduct–performance” paradigm in thefield of industrial organization.

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This evolution of the asset management industry has transformed the ownership structure of firms. Inany industry today, large firms are likely to have common shareholders with significant shares (Azar,Schmalz, and Tecu, 2018).9

Before surveying the evidence on these macroeconomic trends, let us examine what evidence thereis on how common ownership might affect the incentives of managers. Common owners in an industrymay have the ability and incentive to influence management. Indeed, both voice and exit can strengthenwith common ownership (Edmans, Levit, and Reilly, 2019), and not pushing for aggressiveness in man-agement contracts is a mechanism by which common owners can relax competition (Antón, Ederer,Gine, and Schmalz, 2018). Note also that, even if a fund follows a passive strategy and even if a goodpart of the increase in common ownership is due to the rise of passive funds, we cannot assume that thefund is a passive owner (Appel, Gormley, and B.Keim, 2016). In fact, large passive funds tend to exhibita more “disciplinarian” attitude toward management (Bolton, Li, Ravina, and Rosenthal, 2019)—andinstitutional common owners not only internalize governance externalities but also are more likely tovote against management (He, Huang, and Zhao, 2019).10 There are, however, countervailing agencyproblems: Bebchuk and Hirst (2019) point out that index fund managers may not have incentives tomonitor management (for evidence that index funds are less likely to vote against management thanare active funds, see Brav, Jiang, Li, and Pinnington, 2019; Heath, Macciocchi, Michaely, and Ringgen-berg, 2019). Schmidt and Fahlenbrach (2017) show that increased passive ownership impedes high-costgovernance activities and increases agency costs.11 In short, the link between increased passive diversi-fication and relaxed competition may stem either from the internalization by managers of the commonowners’ interests or from increased agency costs that allow managers to slack.12

Recent empirical research has renewed interest in the issue of aggregate market power and its con-sequences for macroeconomic outcomes. Grullon, Larkin, and Michaely (2019) claim that concentra-tion has increased in more than 75% of US industries over the past two decades and also that firmsin industries with larger increases in product market concentration have enjoyed higher profit marginsand positive abnormal stock returns—suggesting that market power is the driver of these outcomes.13

De Loecker, Eeckhout, and Unger (2020) document, for the US economy, a large increase in markups(in excess of the increased overhead) and in economic profits since 1955. These authors attribute thoseincreases to a re-allocation of market share: from low-productivity, low-markup, high–labor share firmsto high-productivity, high-markup, low–labor share firms (in line with the results reported in Autor,

9Minority cross-ownership is also common and has anti-competitive effects (Dietzenbacher, Smid, and Volkerink, 2000;Brito, Osório, Ribeiro, and Vasconcelos, 2018; Nain and Wang, 2018).

10Furthermore, portfolio managers have incentives to increase, even marginally, the value of firms in their portfolio becausedoing so increases management fees (Lewellen and Lewellen, 2018). Jahnke’s (2019) field research, based on 50 interviews withlarge-asset managers, supports the view that they have considerable incentives to engage in corporate governance activitiesfor the purpose of increasing portfolio values. This finding is consistent with the views expressed by large-asset managersthemselves in their “corporate stewardship” reports (e.g., BlackRock, 2019).

11Hansen and Lott (1996) observe that higher agency costs may be associated with more managerial discretion when man-agers internalize externalities through portfolio value maximization.

12Yet when managers hold shares in their firm, agency costs could mitigate the anti-competitive effects of common owner-ship (see Azar, 2020).

13Autor, Dorn, Katz, Patterson, and Van Reenen (2020) state that, for the period 1982-2012, “according to all measures ofsales concentration, industries have become more concentrated on average.”

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Dorn, Katz, Patterson, and Van Reenen, 2020 and Kehrig and Vincent, 2018). Autor, Dorn, Katz, Pat-terson, and Van Reenen (2020) posit that globalization and technological change lead to concentrationand to the rise of what they call “superstar” firms, which have high profits and a low labor share. Asthe importance of superstar firms rises (with the increase in concentration), the aggregate labor sharefalls.14 We find that increased common ownership generates a re-allocation of market share from low-productivity, low-markup, high–labor share firms to high-productivity, high-markup, low–labor sharefirms. There is also substantial evidence that large firms have market power not just in product marketsbut also in labor markets.15 Furthermore, there are claims also of increasing labor market concentration(Benmelech, Bergman, and Kim, forth.).

In addition to increases in concentration as traditionally measured, recent research has shown that:(i) increased overlapping ownership of firms by financial institutions (and by funds in particular)—what we refer to as common ownership—has led to substantial increases in effective (i.e., augmented bycommon ownership) concentration indices in the airline and banking industries; and (ii) this greaterconcentration is associated with higher prices (Azar, Schmalz, and Tecu, 2018). Gutiérrez and Philippon(2017b) suggest that the increase in index and quasi-index fund ownership has played a role in thedecline of aggregate investment.16 Summers (2016) and Stiglitz (2017) link increases in market powerto the potential secular stagnation of developed economies, and Boller and Morton (2020) use an eventstudy of inclusion in the S&P 500 index to conclude that common ownership increases profits.

Some of the recent empirical papers develop theoretical frameworks that link changes in marketpower to the labor share (Eggertsson, Robbins, and Wold, 2018; Barkai, 2020) and to investment and in-terest rates (Brun and González, 2017; Gutiérrez and Philippon, 2017a; Eggertsson, Robbins, and Wold,2018). The models developed by Brun and González (2017), Gutiérrez and Philippon (2017a), Eggerts-son, Robbins, and Wold (2018), and Barkai (2020) are based on a monopolistic competition frameworkwith markups determined exogenously by the parameter reflecting the elasticity of substitution amongproducts. In all cases, only product market power is considered and the firms are assumed to have nomarket power in labor or capital markets. Our theoretical framework differs from these because we ex-plicitly model oligopoly and strategic interaction between firms in general equilibrium, which enablesthe study of how competition policy affects the macro economy. The concern about market power inboth product and labor markets is a subject of policy debate; for example, the Council of EconomicAdvisers produced two reports (CEA, 2016a,b) on the issue of market power. Increased common own-

14Blonigen and Pierce (2016) attribute the US increase in markups to increased merger activity. Barkai (2020) documentsdeclining labor and capital shares in the US economy over the past 30 years, an outcome that is consistent with an increasein markups. Acemoglu and Restrepo (2019), summarizing a body of work, argue that automation always reduces the laborshare in industry value added and that it will tend also to reduce the economy’s overall labor share. For example, Acemogluand Restrepo (2018) report that the labor share declines more in industries (e.g., manufacturing) that are more amenable toautomation.

15A thriving literature in labor economics has established that individual firms face labor supply curves that are imperfectlyelastic, which is indicative of substantial labor market power (Falch, 2010; Ransom and Sims, 2010; Staiger, Spetz, and Phibbs,2010; Matsudaira, 2013; Azar, Marinescu, and Steinbaum, 2020).

16There is an empirical debate on the validity and robustness of these results, since the Modified HHI is endogenous (seeGramlich and Grundl, 2017; Kennedy, O”Brien, Song, and Waehrer, 2017; O’Brien and Waehrer, 2017; Dennis, Gerardi, andSchenone, 2019). Backus, Conlon, and Sinkinson (2018) adopt a structural approach in their study of the cereal industry andfind large potential (but not actual) implied effects of common ownership relative to mergers.

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ership has also raised antitrust concerns (Baker, 2016; Elhauge, 2016) and led to some bold proposals forremedies (Posner, Scott Morton, and Weyl, 2017; Scott Morton and Hovenkamp, 2018) as well as callsfor caution (Rock and Rubinfeld, 2017).

There is an empirical debate about the trends in concentration and markups. Indeed, Rossi-Hansberg,Sarte, and Trachter (2018) find diverging trends for aggregate (increasing) and (decreasing) concentra-tion. Rinz (2018) and Berger, Herkenhoff, and Mongey (2019) find also that local labor market concen-tration has gone down. Traina (2018) and Karabarbounis and Neiman (2019) find flat markups when ac-counting for indirect costs of production. Increases in concentration are modest overall in both productand labor markets and/or on too broadly defined industries to generate severe product market powerproblems (e.g, HHIs remain below antitrust thresholds in relevant product and geographic markets, e.g.Shapiro, 2018).

The question, then, is how to reconcile the evolution of concentration in relevant markets with evi-dence on the evolution of margins, increasing corporate profits, and decreased labor share. Accordingto the monopolistic competition model, margins increase when products become less differentiated. Itis however not plausible that large changes in product differentiation happen in short spans of time. Weprovide an alternative framework in which market concentration and ownership structure both have arole to play.

3 One-sector economy with large firms

In this section we first describe the model in detail. We then characterize the equilibrium and compar-ative statics properties with homogeneous and heterogeneous firms before offering a constant elasticityexample. We conclude with a summary and by describing an extension that allows for investment.

3.1 Model setup

We consider an economy with (a) a finite number of firms, each of them large relative to the economy asa whole, and (b) an infinite number (a continuum) of people, each of them infinitesimal relative to theeconomy as a whole. There are two types of people, workers and owners, and both types consume thegood produced by firms. Workers obtain income to pay for their consumption by offering their time toa firm in exchange for wages. The owners do not work for the firms; an owner’s income derives insteadfrom ownership of the firm’s shares, which entitles the owner to control the firm and to a share of itsprofits. There is a unit mass of workers and a unit mass of owners, and we use IW and IO to denote(respectively) the set of workers and the set of owners. There are a total of J firms in the economy.

There are two goods: a consumer good, with price p; and leisure, with price w. Each worker hasa time endowment of T hours but owns no other assets. Workers have preferences over consumptionand leisure—as represented by the utility function U(Ci, Li), where Ci is worker i’s level of consumptionand Li is i’s labor supply. We assume that the utility function is twice continuously differentiable and

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satisfies UC > 0, UL < 0, UCC < 0, ULL < 0, and UCL ≤ 0.17 The last of these expressions implies thatthe marginal utility of consumption is decreasing in labor supply.

The owners hold all of the firms’ shares. We assume that the owners are divided uniformly into Jgroups, one per firm, with owners in group j owning 1− φ + φ/J of firm j and owning φ/J of the otherfirms; here φ ∈ [0, 1]. Thus φ can be interpreted as representing the level of portfolio diversification, or(quasi-)indexation, in the economy.18

If we use πk to denote the profits of firm k, then the financial wealth of owner i in group j is given by

Wi =1− φ + φ/J

1/Jπj + ∑

k 6=jφπk.

Total financial wealth is equal to ∑Jk=1 πk, the sum of the profits of all firms. The owners obtainutility from consumption only, and for simplicity we assume that their utility function is UO(Ci) = Ci.A firm produces using only labor as a resource, and it has a twice continuously differentiable productionfunction F(L) with F′ > 0 and F(0) ≥ 0. We allow for both F′′ ≤ 0 and F′′ > 0. We use Lj to denote theamount of labor employed by firm j. Firm j’s profits are πj = pF(Lj)− wLj.

We assume that firm j’s objective function is to maximize a weighted average of the (indirect) utilitiesof its owners, where the weights are proportional to the number of shares. In other words, we supposethat ownership confers control in proportion to the shares owned.19 In this simple case, because share-holders do not work and there is only one consumption good, their indirect utility (as a function ofprices, wages, and their wealth level) is VO(p, w; Wi) = Wi/p. Therefore, the objective function of thefirm’s manager is

(1− φ + φ

J

)︸ ︷︷ ︸

Control share ofgroup j in firm j

(1− φ + φJ

)πj +

φJ ∑k 6=j πk

p︸ ︷︷ ︸Indirect utility of shareholder group j

+ ∑k 6=j

φ

J︸︷︷︸Control share ofgroup k in firm j

(1− φ + φJ

)πk +

φJ ∑s 6=k πs

p︸ ︷︷ ︸Indirect utility of shareholder group k

.

After regrouping terms, we can write the objective function as

[(1− φ + φ

J

)2+ (J − 1)

(φ

J

)2]πjp+

[2(

1− φ + φJ

)φ

J+ (J − 2)

(φ

J

)2]∑k 6=j

πkp

.

17Here Ux is the partial derivative of U with respect to variable x, and Uxy is the cross-derivative of U with respect to xand y.

18Each owner in group j is endowed with a fraction (1− φ + φ/J)/(1/J) of firm j and a fraction (φ/J)/(1/J) = φ of each ofthe other firms. Since the mass of the group is 1/J, it follows that the combined ownership in firm j of all the owners in group jis equal to 1− φ + φ/J and that their combined ownership in each of the other firms is φ/J. The combined ownership sharesof all shareholders sum to 1 for every firm:

1− φ + φ/J1/J︸ ︷︷ ︸

Ownership of firm jby an owner in group j

× 1/J︸︷︷︸Mass of group j

+(J − 1)× φ/J1/J︸︷︷︸

Ownership of firm jby an owner in group k 6= j

× 1/J︸︷︷︸Mass of group k

= 1.

19See O’Brien and Salop (2000) for other possibilities that allow for cash flow and control rights to differ.

10

After some algebra we obtain that, for firms’ managers, the objective function simplifies to maximiz-ing (in terms of the consumption good) the sum of own profits and the profits of other firms—discountedby a coefficient λ. Formally, we have

πj

p+ λ ∑

k 6=j

πkp

,

whereλ =

(2− φ)φ(1− φ)2 J + (2− φ)φ .

We interpret λ as the weight—due to common ownership—that each firm’s objective function as-signs to the profits of other firms relative to its own profits. This term was called the coefficient of“effective sympathy” between firms by Edgeworth (1881) and also by Cyert and DeGroot (1973). Theweight λ increases with φ, or the level of portfolio diversification in the economy, and also with marketconcentration 1/J. We remark that λ = 0 if φ = 0 and λ = 1 if φ = 1, so all firms behave “as one” whenportfolios are fully diversified.

Next we define our concept of equilibrium.

3.2 Equilibrium concept

An imperfectly competitive equilibrium with shareholder representation consists of (a) a price function thatassigns consumption good prices to the production plans of firms, (b) an allocation of consumptiongoods, and (c) a set of production plans for firms such that the following statements hold.

(1) The prices and allocation of consumption goods are a competitive equilibrium relative to the pro-duction plans of firms.

(2) Production plans constitute a Cournot–Nash equilibrium when the objective function of each firmis a weighted average of shareholders’ indirect utilities.

It follows then that if a price function, an allocation of consumption goods, and a set of productionplans for firms is an imperfectly competitive equilibrium with shareholder representation, then also ascalar multiple of prices will be an equilibrium with the same allocation of goods and production. Thereason is that (a) the indirect utility function is homogeneous of degree 0 in prices and income; and (b) ifa consumption and production allocation satisfies both (1) and (2) with the original price function, thenit will continue to do so when prices are scaled.

We start by defining a competitive equilibrium relative to the firms’ production plans—in the partic-ular model of this section, a Walrasian equilibrium conditional on the quantities of output announced bythe firms. To simplify notation, we proxy firm j’s production plan by the quantity Lj of labor demanded,implicitly setting the planned production quantity equal to F(Lj).

Definition 1 (Competitive equilibrium relative to production plans). A competitive equilibrium relative to(L1, . . . , LJ) is a price system and allocation [{w, p}; {Ci, Li}i∈IW , {Ci}i∈IO ] such that the following statementshold.

11

(i) For i ∈ IW , (Ci, Li) maximizes U(Ci, Li) subject to pCi ≤ wLi; for i ∈ IO, Ci = Wi/p.

(ii) Labor supply equals labor demand by the firms:∫

i∈IW Li di = ∑Jj=1 Lj.

(iii) Total consumption equals total production:∫

i∈IW∪IO Ci di = ∑Jj=1 F(Lj).

A price function W(L) and P(L) assigns prices {w, p} to each labor (production) plan vector L ≡(L1, . . . , LJ), such that for any L, [W(L), P(L); {Ci, Li}i∈IW , {Ci}i∈IO ] is a competitive equilibrium forsome allocation {{Ci, Li}i∈IW , {Ci}i∈IO}. A given firm makes employment and production plans condi-tional on the price function, which captures how the firm expects prices will react to its plans as wellas its expectations regarding the employment and production plans of other firms. The economy is inequilibrium when every firm’s employment and production plans coincide with the expectations of allother firms.

Definition 2 (Cournot–Walras equilibrium with shareholder representation). A Cournot–Walras equilib-rium with shareholder representation is a price function (W(·), P(·)), an allocation ({C∗i , Li}i∈IW , {C∗i }i∈IO),and a set of production plans L∗ such that:

(i) [W(L∗), P(L∗); {C∗i , Li}i∈IW , {C∗i }i∈IO ] is a competitive equilibrium relative to L∗; and

(ii) the production plan vector L∗ is a pure-strategy Nash equilibrium of a game in which players are the J firms,the strategy space of firm j is [0, T], and the firm’s payoff function is

πj

p+ λ ∑

k 6=j

πkp

.

Here p = P(L), w = W(L), and πj = pF(Lj)− wLj for j = 1, . . . , J.

Note that the objective function of firm j depends only on the real wage ω = w/p, which is invariantto any normalization of prices.

3.3 Characterization of equilibrium

Given firms’ production plans, we derive the real wage—under a competitive equilibrium—by assum-ing that workers maximize their utility U(Ci, Li) subject to the budget constraint Ci ≤ ωLi. This con-straint is always binding because utility increases with consumption but decreases with labor. Substi-tuting the budget constraint into the utility function of the representative worker yields the following(equivalent) maximization problem:

maxLi∈[0,T]

U(ωLi, Li).

Our assumptions on the utility function guarantee that the second-order condition holds. Hence thefirst-order condition for an interior solution implicitly defines a labor supply function h(ω) for worker isuch that labor supply is given by Li = min{h(ω), T}; this coincides with aggregate (average) labor

12

supply, which is∫

i∈I Lidi. Let η denote the elasticity of labor supply. We assume that preferences aresuch that h(·) is increasing.20

Maintained assumption. h′(ω) > 0 for ω ∈ [0, ∞).This assumption is consistent with a wide range of empirical studies showing that the elasticity

of labor supply with respect to wages is positive. A meta-analysis of such studies based on differentmethodologies (Chetty, Guren, Manoli, and Weber, 2011) concludes that the long-run elasticity of ag-gregate hours worked with respect to the real wage is about 0.59. We assume that the range of thelabor supply function is [0, T], which—when combined with the preceding maintained assumption—guarantees the existence of an increasing inverse labor supply function h−1 that assigns a real wage toevery possible labor supply level on [0, T]. In a competitive equilibrium relative to the vector of labordemands by the firms, labor demand has to equal labor supply:

J

∑j=1

Lj =∫

i∈ILidi.

Any competitive equilibrium relative to firms’ production plans L must satisfy either ω = h−1(L) ifL = ∑Jj=1 Lj < T or ω ≥ h−1(T) if L = T.21 In what follows we shall use the price function that assignsω = h−1(T) when L = T. Given that the relative price depends only on L, we can define (with onlyminor abuse of notation) the competitive equilibrium real-wage function ω(L) = h−1(L).

3.4 Cournot–Walras equilibrium: Existence and characterization

Here we identify the conditions under which symmetric equilibria exist. We also offer a characterizationthat relates the markdown of wages (relative to the marginal product of labor) to the economy’s level ofmarket concentration.

The objective of firm j’s manager is to choose an Lj that maximizes the following expression:

F(Lj)−ω(L)Lj + λ ∑k 6=j

[F(Lk)−ω(L)Lk].

We start by noting that firm j’s best response depends only on the aggregate response of its rivals,

∑k 6=j Lk, because the marginal return to firm j is F′(Lj) − ω(L) −(

Lj + λ ∑k 6=j Lk)ω′(L). Let Eω′ ≡

−ω′′L/ω′ denote the elasticity of the inverse labor supply’s slope. Then a sufficient condition for thegame (among firms) to be of the “strategic substitutes” variety is that Eω′ < 1. In this case, one firm’sincrease in labor demand is met by reductions in labor demand by the other firms and so there is anequilibrium (Vives, 1999, Thm. 2.7). Furthermore, if F′′ ≤ 0 and Eω′ < 1 then the objective of the firm is

20We can obtain the slope of h by taking the derivative with respect to the real wage in the first-order condition. Thisprocedure yields

sgn{h′} = sgn{

UC + (UCCω + UCL)∫

i∈ILi di

}.

21The implication here is that the competitive equilibrium real wage as a function of (L1, . . . , LJ) depends on firms’ individ-ual labor demands only through their effect on aggregate labor demand L.

13

strictly concave and the slope of its best response to a rival’s change in labor demand is greater than −1.In that event, the equilibrium is unique (per Vives, 1999, Thm. 2.8).

Proposition 1. Let Eω′ < 1. Then the game among firms is one of strategic substitutes and an equilibrium exists.Moreover, if returns are non-increasing (i.e., if F′′ ≤ 0) then the equilibrium is unique, symmetric, and locallystable under continuous adjustment (unless F′′ = 0 and λ = 1). In an interior symmetric equilibrium, if thetotal employment level L∗ ∈ (0, T) then the following statements hold.

(a) The markdown of the real wage ω∗ is given by

µ ≡ F′(L∗/J)−ω(L∗)

ω(L∗)=

Hη(L∗)

,

where H = (1+λ(J− 1))/J is the modified HHI of the labor market and where H and µ are each increasingin φ.

(b) Both L∗ and ω∗ are increasing in J and decreasing in φ.

(c) The share of a firm’s income received by workers, (ω(L∗)L∗)/(JF(L∗/J)), decreases with φ.

Remark. To ensure a unique equilibrium, it is enough that −F′′(Lj) + (1− λ)ω′(L) > 0 if the second-order condition holds. In this case we may have a unique (and symmetric) equilibrium with moderatelyincreasing returns. Note that F′′ < 0 is required if the condition is to hold for all λ. Furthermore, it ispossible to show that together the inequalities −F′′ + (1− λ)ω′ > 0 and ω′ > 0 are enough to ensurethat a symmetric equilibrium exists and, in addition, that there are no asymmetric equilibria. And ifalso F′′ ≤ 0 and Eω′ < 2 when evaluated at a candidate symmetric equilibrium, then the symmetricequilibrium is unique for any λ (and is stable provided that λ < 1). These relaxed conditions allow forstrategies that are strategic complements.

Remark. If F′′ = 0 (constant returns) and if λ = 1 (φ = 1, firm cartel), then there is a unique symmetricequilibrium and also multiple asymmetric equilibria, with each firm employing an arbitrary amountbetween zero and the monopoly level of employment and with the total employment by firms equal tothat under monopoly. The reason is that the shareholders in this case are indifferent over which firmengages in the actual production.

Remark. The market power friction at a symmetric equilibrium can also be expressed in terms of themarkup of product prices over the effective marginal cost of labor (mc ≡ w/F′(L/JN)),

µ̃ ≡ p−mcp

=µ

1 + µ,

rather than in terms of the markdown

µ =F′ − w/p

w/p=

p−mcmc

.

14

The Lerner-type misalignment of the marginal product of labor and the real wage (i.e., the mark-down µ of real wages) is equal to the modified HHI divided by the elasticity η of labor supply. Thequestion then arises of why there is no effect of the concentration and/or the residual demand elasticityin the product market. In other words: why does there seem to be no effect of product market power? Thereason is that, when there is a single good, this effect (equal to product market modified HHI divided bydemand elasticity) is exactly compensated by the effect of owners internalizing their consumption—thatis, since they are also consumers of the product that the oligopolistic firms produce. Owners use firmprofits only for purchasing the good.22

Additively separable isoelastic preferences and Cobb–Douglas production

We now consider a special case of the model, one in which consumer-workers have separable isoelasticpreferences over consumption and leisure:

U(Ci, Li) =C1−σi1− σ − χ

L1+ξi1 + ξ

,

where σ ∈ (0, 1) and χ, ξ > 0. The elasticity of labor supply is η = (1 − σ)/(ξ + σ) > 0, and theequilibrium real wage in the competitive equilibrium—given firms’ aggregate labor demand—can bewritten as

ω(L) = χ1/(1−σ)L1/η

with elasticitiesω′Lω

=1η

and Eω′ = 1−1η< 1.

The production function is F(Lj) = ALαj , where A > 0, 0 < α ≤ 1, and returns are non-increasing.The objective function of each firm is strictly concave, and so Proposition 1 applies. It is easily

checked that total employment under the unique symmetric equilibrium is

L∗ =(

χ−1/(1−σ) J1−αAα

1 + H/η

)1/(1−α+1/η).

Figure 1 illustrates that an increase in common ownership—that is, an increase in φ or a decreasein the number of firms—reduces equilibrium employment and real wages. With increasing returns toscale, however, reducing the number of firms involves a trade-off between market power and efficiency.In that case, a decline in the number of firms can increase real wages under some conditions.

[[ INSERT Figure 1 about Here ]]

The symmetric equilibrium is locally stable if α− 1 < (1− λ)(Jη)−1(1 + H/η)−1, which means that22In contrast to the partial equilibrium model of Farrell (1985), the equilibrium markdown in our model—because of the labor

market power effect—is not zero even when ownership is proportional to consumption. If the labor market is competitive (i.e.,if η = ∞) then the equilibrium markdown is zero. See also Mas-Colell and Silvestre (1991).

15

a range of increasing returns may be allowed provided that an equilibrium exists. If α > 1, then neitherthe inequality−F′′+ (1− λ)ω′ > 0 nor the payoff global concavity condition need hold. In Appendix Bwe characterize the case where α ∈ (1, 2) and η ≤ 1 and then give a necessary and sufficient conditionfor an interior symmetric equilibrium to exist when returns are increasing. Under that condition, L∗ isdecreasing in φ; yet it may either increase or decrease with J depending on whether the effect on themarkdown or the economies of scale prevail.

3.5 Heterogenous firms

When firms have access to different constant returns to scale technologies (CRS), we confirm the resultsin Proposition 1 and establish a positive association between common ownership and the dispersion ofmarket shares.

Proposition 2. Let Eω′ < 1 and firms have potentially different CRS technologies with Fj(Lj) = AjLj, Aj > 0,j = 1, . . . , J. Then an equilibrium exists and is unique with λ < 1. In an interior equilibrium with L∗ ∈ (0, T),the following statements hold.

(a) The markdown of real wages for firm j is given by

µj ≡F′j (L

∗j )−ω(L∗)ω(L∗)

=s∗j + λ(1− s∗j )

η(L∗),

where s∗j ≡ L∗j /L∗ and where the weighted average markdown is

µ̄ ≡J

∑j=1

sjµj =H

η(L∗);

here H = HHI(1− λ) + λ is the modified HHI, and both H and µ̄ are increasing in φ.

(b) The total employment level L∗ and the real wage ω∗ are each decreasing in φ.

(c) The share of income going to workers, (ω(L∗)L∗)/(JF(L∗/J)) = 1/(1 + µ̄), decreases with φ.

(d) If technologies are heterogeneous, then: (a) both the HHI and the minimal relative productivity for the leastproductive firm to be viable (i.e., Amin/Ā, where Ā = ∑

Jj=1 Aj/J) are increasing in φ; and (b) only the

most productive firm is active when φ→ 1.

Thus, under firm heterogeneity, an increase in common ownership as measured by φ (and λ) leadsendogenously to an increase in the Herfindahl–Hirschman index. This follows because a firm’s marketshare sj increases (resp. decreases) when λ increases if j has above-average (resp. below-average) pro-ductivity. The implication is that, when common ownership increases, the variance of sj increases andso the HHI increases as well. The effect of common ownership is similar to the behavior of a multi-plant

16

monopolist who shifts production towards the more efficient plants.23 Common ownership thus gener-ates a re-allocation of market share from low-productivity, low-markup, high–labor share firms to high-productivity, high-markup, low–labor share firms. As stated in Section 2.2, a pattern of re-allocationfrom low- to high-markup US firms in recent decades is documented by De Loecker, Eeckhout, andUnger (2020), and Autor, Dorn, Katz, Patterson, and Van Reenen (2020) and Kehrig and Vincent (2018)both find evidence of a re-allocation from high–labor share to low–labor share firms.

Remark. At a Cournot interior equilibrium with constant marginal costs, total output does not dependon the distribution of costs (e.g., Bergstrom and Varian, 1985). Here, too, we have that total employmentdepends only on average productivity Ā = ∑Jj=1 Aj/J and not on its variance. However, a technologicalchange that induces a discrete increase in the dispersion of productivities, large enough to induce theexit of inefficient firms from the market, does affect equilibrium employment. In addition, an increasein common ownership may reinforce this effect. Indeed, we can show that the minimal relative produc-tivity for the least productive firm to be viable (Amin/Ā) is given by

(η+λ)/(1−λ)(η+λ)/(1−λ)+1/J , which is increasing

in λ.24 Moreover, if it is not profitable for the jth least productive firm to produce a positive amount inequilibrium, then it is also not profitable to produce with λ′ > λ. As φ → 1, only the most productivefirm survives and behaves like a monopsonist.

3.6 Summary and investment extension

So far we have shown that the simple model developed in this section can help make sense of some re-cent macroeconomic stylized facts—including persistently low output, employment, and wages in thepresence of high corporate profits and financial wealth—as a response to a permanent increase in effec-tive concentration (due either to common ownership or to a reduced number of competitors). Becausewe have yet to incorporate investment decisions into the model, there is no real interest rate and sowe have nothing to say about how it is affected. Even so, the model can be extended to include sav-ing, capital, investment, and the real interest rate. In Azar and Vives (2019a) we present a model withworkers, owners and savers and show that—for investors who are not fully diversified—either a fallin the number J of firms or a rise in φ, the common ownership parameter, will lead to an equilibriumwith lower levels of capital stock, employment, real interest rate, real wages, output, and labor share ofincome. Under certain (reasonable) conditions, the changes just described will lead also to a decliningcapital share.

When firms are large relative to the economy, an increase in market power implies that firms havean incentive to reduce both their employment and investment below the competitive level; this followsbecause, even though such firms sacrifice in terms of output, they benefit from lower wages and lowerinterest rates on every unit of labor and capital that they employ. The effect described here is present onlywhen firms’ shareholders perceive that they can affect the economy’s equilibrium level of real wages and

23It follows that increases in common ownership will raise the relative incentives of the more efficient firms to invest in costreduction—that is, since they will end up producing more (see the model in López and Vives, 2019).

24To see that the threshold is increasing in λ we use Lemma 1 in Appendix B, which shows that in equilibrium ∂η∂λ + 1 > 0.

17

real interest rates by changing their production plans. Thus, when oligopolistic firms have market powerover the economy as a whole, their owners can extract rents from both workers and savers.25

4 Multiple sectors

In this section we extend the model to multiple sectors in a Cobb–Douglas isoelastic environment. Wecharacterize the equilibrium, uncover new and richer comparative statics results, and proceed to analyzelarge markets and convergence to the monopolistic competition outcome as the number of sectors growslarge. We end the section with a note on calibration of the model.

4.1 Model setup

Consider an economy with N sectors, each offering a different consumer product. We assume that boththe mass of workers and the mass of owners are equal to N. So as we scale the economy by increasingthe number of sectors, the number of people in the economy scales proportionally. The utility functionof worker i is as in the additively separable isoelastic model: U(Ci, Li) = C1−σi /(1− σ)− χL

1+ξi /(1 + ξ)

for σ ∈ (0, 1) and χ, ξ > 0, where

Ci =[(

1N

)1/θ N∑n=1

c(θ−1)/θni

]θ/(θ−1);

here cni is the consumption of worker i in sector n, and θ > 1 is the elasticity of substitution indicating apreference for variety.26

For each product, there are J firms that can produce it using labor as input. The profits of firm j insector n are given by

πnj = pnF(Lnj)− wLnj;

here, the production function is F(Lnj) = ALαnj with A > 0 and α > 0.The ownership structure is similar to the single-sector case, except that now (i) there are J×N groups

of shareholders and (ii) shareholders can diversify both in an industry fund and in a economy-widefund. Group nj owns a fraction 1− φ− φ̃ ≥ 0 in firm nj directly, an industry index fund with a frac-tion φ̃/J in every firm in sector n, and an economy-wide index fund with a fraction φ/NJ in every firm.The owners’ utility is simply their consumption Ci of the composite good. Solving the owners’ util-ity maximization problem yields the indirect utility function of shareholder i (i.e., V(P, w; Wi) = Wi/P)when prices are {pn}Nn=1, the level of wages is w, shareholder wealth is Wi, and P ≡

( 1N ∑

Nn=1 p1−θn

)1/(1−θ)is the price index.

25Our model does not account for the possibility of inter-firm technological spillovers due to investment. López and Vives(2019) show that, if spillovers are high enough, then increased common ownership may boost R&D investment as well.

26The form of Ci is the one used by Allen and Arkolakis (2016). The weight (1/N)1/θ in Ci implies that, as N grows, theindirect utility derived from Ci does not grow unboundedly and is consistent with a continuum formulation for the sectors(replacing the summation with an integral) of unit mass. More precisely: if the equilibrium is symmetric then, regardless of N,the level of consumption Ci is equal to the consumer’s income divided by the price.

18

The objective function of the manager of firm j in sector n is to choose the firm’s level of employment,Lnj, that maximizes a weighted average of shareholder (indirect) utilities. By re-arranging coefficientsso that the coefficient for own profits equals 1, we obtain the objective function

πnj

P︸︷︷︸own profits

+λintra ∑k 6=j

πnkP︸ ︷︷ ︸

industry n profits, other firms

+λinter ∑m 6=n

J

∑k=1

πmkP︸ ︷︷ ︸

profits, other industries

,

where the lambdas are a function of (φ, φ̃, J, N).Thus the firm accounts for the effects of its actions not only on same-sector rivals but also on firms

in other sectors. Note that the manager’s objective function depends on N + 1 relative prices—that is,on w/P in addition to {pn/P}Nn=1 for N > 1.

We can show that the Edgeworth sympathy coefficient for other firms in the same sector as the focalfirm is

λintra =(2− φ)φ + [2(1− φ)− φ̃]φ̃N

(1− φ)2 JN + (2− φ)φ− [2(1− φ)− φ̃]φ̃N(J − 1)

and that the Edgeworth sympathy coefficient for firms in other sectors is given by

λinter =(2− φ)φ

(1− φ)2 JN + (2− φ)φ− [2(1− φ)− φ̃]φ̃N(J − 1).

Observe that λintra is no less than λinter. This follows because the former sums the profit weights of boththe industry fund and the economy-wide fund. We can show (see Lemma 2 in Appendix B) that λintraand λinter are always in [0, 1], increasing in φ and φ̃, and—for φ > 0 and φ + φ̃ < 1—decreasing in Nand J.

When φ + φ̃ = 1, we have λintra = 1 and λinter = (1− φ̃2)/[1 + φ̃2(N − 1)]; as a result, if agents arefully invested in the two index funds then λintra = 1 regardless of the share in each fund. In contrast,the sympathy λinter for firms in other sectors decreases as shares are moved from the economy indexfund to the own-industry index fund φ̃.27 Indeed, if everything is invested in the industry fund thenφ̃ = 1, λintra = 1, and λinter = 0. If there is no economy-wide index fund, then φ = 0, λinter = 0, andλintra =

(2−φ̃)φ̃(1−φ̃)2 J+(2−φ̃)φ̃ . If there are no industry funds then φ̃ = 0 and λintra = λinter =

(2−φ)φ(1−φ)2 JN+(2−φ)φ .

28

Finally, if everything is invested in the economy-wide index fund then φ = 1 and λintra = λinter = 1.

4.2 Cournot–Walras equilibrium with N sectors

We start by characterizing the competitive equilibrium in terms of relative prices w/P and of {pn/P}Nn=1,given the production plans of the J firms operating in the N sectors: L ≡ {L1, . . . , LJ}, where Lj ≡(L1j, . . . , LNj). Then we characterize the equilibrium in the plans of the firms.

27When φ + φ̃ = 1, two firms in the same industry have the same ownership structure, each with φ and φ̃ proportions ofeach fund. Therefore, there is shareholder unanimity in maximizing joint industry profits and λintra = 1.

28In both cases, λ is given as in the one-sector economy: in the first case with φ̃ instead of φ, and in the second with JNinstead of J.

19

4.2.1 Relative prices in a competitive equilibrium given firms’ production plans

Because the function that aggregates the consumption of all sectors is homothetic, workers face a two-stage budgeting problem. First, workers choose their consumption across sectors (conditional on theiraggregate level of consumption) to minimize expenditures; second, they choose labor supply Li andconsumption level Ci to maximize their utility U(Ci, Li) subject to the budget constraint PCi = wLi,where P is the price index.

We can therefore write the first-stage problem as

min{cni}Nn=1

N

∑n=1

pncni

subject to [ N∑n=1

(1N

)1/θc(θ−1)/θni

]θ/(θ−1)= Ci.

The solution to this problem yields the standard demand of consumer i for each product n conditionalon aggregate consumption Ci:

cni =1N

(pnP

)−θCi. (4.2.1)

It follows from homotheticity that, for every consumer, total expenditures equal the price index multi-plied by their respective level of consumption:

N

∑n=1

pncni = PCi.

In the second stage, the first-order condition for an interior solution is given by

wP

= −UL(w

P Li, Li)

UC(w

P Li, Li) . (4.2.2)

Since workers are homogeneous, it follows that total labor supply∫

i∈I Li di is simply N times the indi-vidual labor supply Li; moreover, because total labor demand L must equal total labor supply, equa-tion (4.2.2) implicitly defines the equilibrium real wage (now relative to the price of the composite good)as a function ω(L) of the firms’ total employment plans. We retain the assumptions for increasing laborsupply that ensure ω′ > 0. Then ω(L) = χ1/(1−σ)(L/N)1/η , where again η = (1− σ)/(ξ + σ) is theelasticity of labor supply.

Shareholders maximize their aggregate consumption level conditional on their income. Their con-sumer demands, conditional on their respective levels of consumption, are identical to those of workers.

20

Adding up the demands across owners and workers, we obtain

∫i∈IW∪IO

cni di︸ ︷︷ ︸cn

=1N

(pnP

)−θ ∫i∈IW∪IO

Ci di︸ ︷︷ ︸C

.

In a competitive equilibrium, consumption demand must equal the sum of all firms’ production of eachproduct:

cn =J

∑j=1

F(Lnj). (4.2.3)

Using equation (4.2.1) and integrating across consumers, we have that cn = 1N( pn

P

)−θC. So givenfirms’ production plans, the following equality holds in a competitive equilibrium:

pnP

=

(1N

)1/θ( cnC

)−1/θ. (4.2.4)

The elasticity of the relative price of sector n, pn/P, with respect to the aggregate production cn ofthe sector for given production in the other sectors (cm for m 6= n), when evaluated at a symmetricequilibrium, is −(1− 1/N)/θ. Its absolute value is decreasing in the elasticity of substitution of thevarieties (θ) and increasing in the number of sectors (N). Increasing cn has a direct negative impacton pn/P of −1/θ for a given C, and an indirect positive impact on pn/P by increasing aggregate realincome C, yielding 1/(θN). When there is only one sector (N = 1) there is obviously no impact on therelative price. Furthermore, the overall effect increases with the number N of sectors because then theindirect effect is weaker.

We can now use equations (4.2.3) and (4.2.4) to obtain an expression for ρn ≡ pn/P in a competitiveequilibrium conditional on firms’ production plans L:

ρn(L) =(

1N

)1/θ{ ∑Jj=1 F(Lnj)[∑Nm=1

( 1N

)1/θ(∑Jj=1 F(Lmj)

)(θ−1)/θ]θ/(θ−1)}−1/θ

.

Observe that—unlike the previous case of a real-wage function, where the dependence was only throughtotal employment plans—relative prices under a competitive equilibrium depend directly on the em-ployment plans of each individual firm.

Proposition 3. Given the production plans L ≡ {Lmj} of firms with aggregate labor demand L, the competitiveequilibrium is given by the real wage ω(L) and the relative prices of the N sectors: ρn(L) for n = 1, . . . , N. Iffirm j in sector n expands its employment plans, then ω increases; in addition, ρn decreases (∂ρn/∂Lnj < 0) whileρm, m 6= n, increases (∂ρm/∂Lnj > 0).

An increase in employment by a firm in sector n increases the relative supply of the consumptiongood of that sector relative to other sectors, thereby reducing the relative price of the focal sector’s

21

good. Since this increased employment increases overall supply of the aggregate consumption goodwhile leaving supply of the other sectors unchanged, the relative prices of goods in those other sectorsincrease.

4.2.2 Cournot–Walras equilibrium

The optimization problem of firm j in sector n is given by

maxLnj

{πnj

P︸︷︷︸own profits

+λ ∑k 6=j

πnkP︸ ︷︷ ︸

industry n profits, other firms

+λ ∑m 6=n

J

∑k=1

πmkP︸ ︷︷ ︸

profits, other industries

},

where πnj/P = ρnF(Lnj)−ω(L)Lnj. The first-order condition for the firm is

ρn (L) F′(

Lnj)︸ ︷︷ ︸

VMPL

− ω (L)︸ ︷︷ ︸real wage

− ∂ω∂Lnj(+)

[Lnj + λ ∑

k 6=jLnk + λ ∑

m 6=n

J

∑k=1

Lmk

]︸ ︷︷ ︸

(i) wage effect

+∂ρn∂Lnj(−)

[F(

Lnj)+ λ ∑

k 6=jF (Lnk)

]︸ ︷︷ ︸

(ii) own-industry relative price effect

+ λ ∑m 6=n

∂ρm∂Lnj(+)

[J

∑k=1

F (Lmk)

]︸ ︷︷ ︸

(iii) other industries’ relative price effect

= 0

When a firm in a given sector considers hiring an additional worker, it faces the following trade-offs. On the one hand, expanding employment increases profits by the value of the marginal product oflabor (VMPL), which the shareholders can consume after paying the new workers the real wage. On theother hand, expanding employment will increase real wages for all workers because the labor supplyis upward sloping. So when there is common ownership, the owners will take into account the wageeffect not just for the firm that expans employment (or just for the firms in the same industry) but forall firms in all industries. Furthermore, expanding employment will increase output in the firm’s sectorand thereby reduce that sector’s relative prices; as before, owners internalize that reduction not just forthe firm itself but for all firms in the sector in which they have common ownership. Finally, expandingoutput in the firm’s sector decreases consumption in all the other sectors and thus increases their relativeprices; the owners of the firm, if they have common ownership involving other sectors, internalize theseincreased relative prices as a positive pecuniary externality. However, we will show that the own-sectornegative price effect always dominates the effect of increased demand in other sectors.

As we establish in Appendix B, a firm’s objective function is strictly concave if α ≤ 1. We thereforehave the following existence and characterization result.29

29As in the one-sector case, if φ = 1 and α = 1 then there is a unique symmetric equilibrium and there also exist asymmetricequilibria, since shareholders are indifferent to which firms employ the workers as long as total employment is at the monopolylevel.

22

Proposition 4. Consider a multi-sector economy with additive separable isoelastic preferences and a Cobb–Douglas production function under non-increasing returns to scale (α ≤ 1). There exists a unique symmetricequilibrium, and equilibrium employment is given by

L∗ = N(

J1−αχ−1/(1−σ)Aα

1 + µ∗

)1/(1−α+1/η).

The equilibrium markdown of real wages is

µ∗ =1 + Hlabor/η

1− (Hproduct − λinter)(1− 1/N)/θ− 1,

where Hlabor ≡ (1+ λintra(J− 1) + λinter(N− 1)J)/NJ is the modified HHI of the labor market and Hproduct ≡(1 + λintra(J − 1))/J is the modified HHI for each sector.

The markdown µ∗ decreases with J (for φ+ φ̃ < 1, with µ∗ → 0 as J → ∞), η, and θ (for φ < 1); it increaseswith φ̃; and it can be non-monotone in φ.

If φ̃ = 0 (no industry fund, λintra = λinter = λ), then Hproduct − λ = (1− λ)/J and

sgn{

∂µ∗

∂φ

}= sgn

{θ

1 + η− N − 1

JN − 1

}.

Remark. Simulations reveal that µ∗ may be non-monotone in φ also if φ̃ > 0. In fact, we can show that if ηis large enough then µ∗ is decreasing in φ for φ̃ > 0 small and increasing in φ for JN large. Furthermore,µ∗ is found to be either increasing or decreasing in N.

In the multiple-industry case we find that the equilibrium real wage, employment, and output areanalogous—as a function of the markdown—to those in the single-industry case. The only differenceis that the markdown is now more complicated owing to the existence of multiple sectors and of prod-uct differentiation across firms in different sectors. An important result that contrasts with the single-sector case is that employment, output, and the real wage may all increase with diversification using theeconomy-wide fund φ.

Perfect substitutes. As the elasticity of substitution (θ) tends to infinity, the products of the differentsectors become close to perfect substitutes; then the equilibrium is just as in the one-industry case butwith JN firms instead of J firms. This outcome should not be surprising given that, in the case of perfectsubstitutes, all firms produce the same good and so—for all intents and purposes—there is but a singleindustry in the economy.

The two wedges of the markdown. The markdown of wages below the marginal product of la-bor can be viewed as consisting of two “wedges”, one reflecting labor market power and one reflectingproduct market power. In particular, the labor market wedge is 1 + Hlabor/η. The markdown is increas-ing in Hlabor/η, which reflects the level of labor market power (and so decreases with JN and η). Theproduct market wedge is (Hproduct − λinter)(1 − 1/N)/θ. This wedge has two components: the first isHproduct(1− 1/N)/θ, reflecting the level of market power in the firm’s sector; the second is λinter(1−

23

1/N)/θ, reflecting the inter-sectoral externality (note that the latter diminishes as products become moresubstitutable and θ increases). The markdown is increasing in the first component of the product marketwedge, and decreasing in the second component.30

From the previous paragraph it follows that µ∗ is positively associated with λintra because so also areboth the labor and product wedges—that is, since Hlabor and Hproduct are increasing in λintra. However,µ∗ may be positively or negatively associated with λinter because, when λinter > 0, we must accountfor the effect of expanding employment (by firm j in sector n) on the profits of other firms. Expandingemployment in one sector benefits firms in other sectors by increasing the relative prices in those sectors(pecuniary externality) via the increase in overall consumption generated by firm nj’s expanded em-ployment plans. The result is that Hproduct is then reduced by λinter (note that Hproduct ≥ λinter always).If an increase in λinter increases the labor market wedge more than it reduces the product market wedge,then µ∗ is decreasing in λinter; the converse of this statement holds as well.

Case with no industry fund. When φ̃ = 0, we have λintra = λinter = λ; then the net effect of anincrease in λ (due to an increase in φ) will be to diminish the product market wedge. To see this, notethat (Hproduct − λ)(1− 1/N)/θ = (1− λ)(1− 1/N)/(θ J). In the limit, when φ = 1 and λ = 1, we havea cartel or a monopoly and the two product market effects cancel each other out exactly.31 It is worthnoting that µ∗ may either increase or decrease with portfolio diversification φ depending on whetherlabor market effects or rather product market effects prevail. The markdown will be decreasing in φwhen the increase in the labor market wedge (due to the higher φ) is more than compensated by thelower product market wedge (due to the pro-competitive inter-sectoral pecuniary externality)—in otherwords, when the effect of profit internalization on the level of market power in product markets is higherthan it is in the labor market. This happens when the elasticity of substitution θ is small in relation tothe elasticity of labor supply η. When η → ∞, common ownership always has a pro-competitive effect.If N is large, then the anti-competitive effect of common ownership prevails provided that η < 1. Thisoutcome follows because then θ/(1 + η) > 1/2 and

sgn{

∂µ∗

∂φ

}= sgn

{θ

1 + η− 1

J

}> 0.

Under the parameter configurations for the elasticities considered in Azar and Vives (2019a), θ = 3 andwith a conservative η = 0.6, we have that θ1+η >

12 . In consequence, the anticompetitive effect will

prevail for N large.

30Recall that, when evaluated at a symmetric equilibrium, the (absolute value of the) elasticity of “inverse demand” pn/Pwith respect to cn is (1− 1/N)/θ; this explains why Hproduct(1− 1/N)/θ is the indicator of product market power (note thatthis indicator decreases with J and θ but increases with N).

31When portfolios are perfectly diversified (φ = 1), the economy can be viewed as consisting of a single large firm thatproduces the composite good. Since the owner-consumers own shares in each of the components of the composite good inthe same proportion and since they use profits only to purchase that good, these owner-consumers are to the same extentshareholders and consumers of the composite good. So just as in the single-sector economy, the effects cancel out exactly. TheN = 1 case is the one-sector model developed in Section 3. Here λ = 1 can be understood in similar terms except that, in thiscase, there is an aggregate good C.

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4.3 Large economies

Most of the literature on oligopoly in general equilibrium considers the case of an infinite number ofsectors such that each sector, and therefore each firm, is small relative to the economy. Monopolisticcompetition can be viewed as a special case of a model with infinite sectors in which there is only onefirm per industry. Here we consider what happens when the number of sectors, N, tends to infinity.Our aim is to identify the conditions under which the monopolistically competitive limit is obtained (asin 1977monopolistic). We consider the following cases where owners: (i) hold fully diversified portfo-lios, (ii) are not diversified (iii) are fully diversified only intra-industry for N large, and (iv) are fullydiversified for N large.

4.3.1 Case 1: Full diversification (φ = 1, φ̃ = 0)

When all the owner-consumers hold market portfolios, λintra(N) = λinter(N) = 1. This means that wehave a sequence of economies with an increasing number of sectors and firms but in which the equilib-rium outcome remains the same. The product market wedge disappears because the owner-consumersfully internalize the effect of firms’ decisions on themselves as consumers. As with the one-sector model,the labor market wedge remains at the monopsony level—here, because owner-consumers still have anincentive to reduce the real wages of worker-consumers. We remark that, if the model had a represen-tative agent rather than owner-consumers and worker-consumers, then the labor market wedge wouldalso disappear and the equilibrium would be efficient.

4.3.2 Case 2: No diversification (φ = φ̃ = 0)

Consider now the case in which owner-consumers hold shares in only one firm. In this case, as thenumber of sectors tends to infinity, the labor market wedge disappears as the number of firms interactingin the labor market goes to infinity (this result would not hold if the labor markets were segmented, forexample, by industry). With J > 1 firms, the limit economy is equivalent to that of Neary (2003b): acontinuum of sectors, no labor market power, and a homogeneous-goods Cournot equilibrium in eachsector (if goods were heterogeneous within sector, then the limit economy would be equivalent to thatof Atkeson and Burstein, 2008). In the case of J = 1, the limit economy in this case is equivalent to thatof the Dixit and Stiglitz (1977) monopolistic competition model.

One must bear in mind, however, that obtaining these economies as a limit in the model requiresheterogeneous agents: owner-consumers and worker-consumers; also, within the owner-consumers,there must be different groups with each group having ownership in just one firm. If, as in Neary(2003b), Atkeson and Burstein (2008), and Dixit and Stiglitz (1977), we assumed a representative agent(a) there would be fully diversified owner-worker-consumers; and (b) the equilibrium of the economyat each point in the sequence of economies would be efficient, with price equal to marginal cost. Eventhough the models in these papers assume profit maximization, no shareholder would actually wantthe firms to maximize profits. This tension was discussed in Section 2.

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4.3.3 Case 3: Only intra-industry asymptotic diversification (φ = 0, φ̃N → φ̃ > 0)

When φ = 0, oligopsony power vanishes in the limit because (again) the number of firms competing inthe labor market goes to infinity and there is no inter-industry internalization effect (λinter(N) = 0). Thus,the limit economy is equivalent to that of Neary (2003b) but with horizontal, within-industry commonownership. In this case, for any N we have the same formula as for the one-sector model except with φ̃Ninstead of φ:

λintra(N) =(2− φ̃N)φ̃N

(1− φ̃N)2 J + (2− φ̃N)φ̃N.

Here the markdownµ∗N → µ∗∞ =

11− Hproduct(∞)/θ

− 1,

increases with φ̃ when J > 1 (in this formula, Hproduct(∞) refers to the limit product market modifiedHHI, which is 1/J + λintra(∞)(1− 1/J)). If φ̃ = 1, then Hproduct(∞) = 1 and µ∗∞ = 1/(θ − 1).

Recall that the market power friction at a symmetric equilibrium can also be expressed in terms ofthe markup of product prices over the effective marginal cost of labor (mc ≡ w/F′(L/JN)),

µ̃ ≡ p−mcp

=µ

1 + µ,

rather than in terms of the markdown. We have that µ̃∗ → 1/θ (the monopolistic competition markupof Dixit and Stiglitz, 1977) when there is essentially one firm per sector (either J = 1 or λintra(∞) = 1; e.g.,φ̃→ 1).32

4.3.4 Case 4: Full asymptotic diversification (φN → 1, φ̃ = 0)

We consider now the case with full asymptotic diversification (φN → 1). For simplicity, we assume noindustry fund: φ̃ = 0, where λintra(N) = λinter(N) = λN =

(2−φN)φN(1−φN)2 JN+(2−φN)φN

. We start by observing that,if φN → φ < 1, then λN → 0. This is so because, as the number of sectors in the economy increases: fora shareholder in group nj, the fraction held in each of the other firms (when φ is constant) is φ/(NJ),which goes to zero, while the fraction 1− φ + φ/(NJ) held in firm nj does not. In this case, then, theequilibrium of the limit economy is like the one in Case 2 (no diversification); hence it is equivalent toNeary (2003b) when J > 1 and to Dixit and Stiglitz (1977) when J = 1.

Consider now the case when φN → 1, or, equivalently, 1− φN → 0. In that case, the limit lambdascan take values between zero and one, depending on the speed of convergence. In particular, to have

32Similar results are obtained with some economy-wide diversification. Suppose φ < 1 and φ̃ > 0 are fixed; then, as N → ∞,we have that λinter → 0 (and oligopsony power vanishes, since Hlabor → λinter(∞) = 0) but that

λintra → λintra(∞) ≡2γ− 1

γ2 J − (2γ− 1)(J − 1) ,

where γ ≡ (1− φ)/φ̃ > 0 is the ratio of undiversified investment to investment in the industry fund. The parameter γ rangesfrom 1 to infinity: γ = 1 when 1− φ = φ̃ (e.g., as when φ̃ = 1); and γ→ ∞ as φ̃→ 0. If γ = 1 then λintra(∞) = 1, and if γ→ ∞then λintra(∞) = 0.

26

λN → λ ∈ (0, 1], we need the sequence 1− φN to approach zero (full diversification) at least as rapidlyas 1/

√N (i.e.,

√N(1− φN) → k for k ∈ [0, ∞)). If the convergence rate is faster than 1/

√N with k = 0,

then the limiting λ is always equal to 1, and the equilibrium in the limit economy is the same as in Case 1(full diversification). If the convergence rate is slower than than 1/

√N then the limiting λ is equal to

zero, and the equilibrium in the limit economy is the same as in Case 2 (no diversification).For sequences 1−φN with convergence rates equal to 1/

√N, the value of λ in the limit is determined

by k, the constant of convergence: if√

N(1− φN) → k then limN→∞ λN = 1/(1 + Jk2).33 If λN → λ∞,then the limit markdown is

µ∗∞ ≡ limN→∞ µ∗N =

1 + λ∞/η1− (1− λ∞)/(θ J)

− 1.

The impact of λ∞ on the markdown depends, as before, on whether (or not) its effect on the labormarket wedge effect dominates its effect on the product market wedge. The labor market wedge effectdominates the product market wedge effect if and only if the elasticity η of labor supply is lower thanθ J − 1. These results are summarized in our next proposition.

Proposition 5. Consider a sequence of economies (φ̃N , φN , N), where φ̃N = 0 for all N but attaining full diver-sification as φN → 1. If

√N(1− φN)→ k for k ∈ [0, ∞) then, as N → ∞, we have λN → 1/(